Using Ampere's circuital law, derive an expression for the magnetic field along the axis of current carrying torodial solenoid of N number of turns having radius r.

Magnetic field due to a toroidal solenoid : 

A long solenoid in the the form of a circular ring is known as torroidal solenoid.

Let,
number of turns pr unit length of the solenoid be 'n', 
I - Current flowing throught the coil, 
B- magnetic field inside the turns of the solenoid. 

Magnetic lines of force inside the torroid is in the form of concentric circles. Using the law of symmetry, the magnetic field is along the tangent at every point on the circle and is same at each point of the circle.

For points inside the core of toroid, 
Let us consider a circle of radius 'r'.
Length of torid = circumference of circular path = 2$\mathrm{\pi }$r 
Total no. of turns in torroid = n. (2$\pi$r ) 
Current flowing across each turn = I 
Therefore,
Total current enclosed by circular path = n (2$\pi$r) I 
Now, using Ampere's circuital law, 
                    $\oint B. dl = {\mu }_{o }I$ 
and       $\oint B. dl \cos 0 = B . 2\pi r$ 

Substituting this in Ampere's law we get, 

            $$\begin{gathered} \mathrm{B}. 2\mathrm{\pi r} = {\mathrm{\mu }}_{\mathrm{O} }( \mathrm{n} . 2\mathrm{\pi r}. \mathrm{I}) \\ \mathrm{B} = {\mathrm{\mu }}_{\mathrm{O} } \mathrm{n} \mathrm{I} \end{gathered}$$ 

which is the required magnetic field along the axis of current carrying toroidal solenoid.